# HG changeset patch
# User wenzelm
# Date 1395326329 -3600
# Node ID 0f6dc7512023496ee3a74dfc3be166ed3f14b9fd
# Parent  00112abe9b256c9343ac82fa76cea7275cd78b05
tuned proofs;

diff -r 00112abe9b25 -r 0f6dc7512023 src/HOL/Library/Lattice_Algebras.thy
--- a/src/HOL/Library/Lattice_Algebras.thy	Thu Mar 20 12:43:48 2014 +0000
+++ b/src/HOL/Library/Lattice_Algebras.thy	Thu Mar 20 15:38:49 2014 +0100
@@ -18,9 +18,10 @@
 
 lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"
 proof -
-  have "c + inf a b = inf (c+a) (c+b)"
+  have "c + inf a b = inf (c + a) (c + b)"
     by (simp add: add_inf_distrib_left)
-  thus ?thesis by (simp add: add_commute)
+  then show ?thesis
+    by (simp add: add_commute)
 qed
 
 end
@@ -37,10 +38,12 @@
   apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
   done
 
-lemma add_sup_distrib_right: "sup a b + c = sup (a+c) (b+c)"
+lemma add_sup_distrib_right: "sup a b + c = sup (a + c) (b + c)"
 proof -
-  have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
-  thus ?thesis by (simp add: add_commute)
+  have "c + sup a b = sup (c+a) (c+b)"
+    by (simp add: add_sup_distrib_left)
+  then show ?thesis
+    by (simp add: add_commute)
 qed
 
 end
@@ -54,10 +57,10 @@
 lemmas add_sup_inf_distribs =
   add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
 
-lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
+lemma inf_eq_neg_sup: "inf a b = - sup (- a) (- b)"
 proof (rule inf_unique)
   fix a b c :: 'a
-  show "- sup (-a) (-b) \<le> a"
+  show "- sup (- a) (- b) \<le> a"
     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
       (simp, simp add: add_sup_distrib_left)
   show "- sup (-a) (-b) \<le> b"
@@ -68,26 +71,27 @@
     by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)
 qed
 
-lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
+lemma sup_eq_neg_inf: "sup a b = - inf (- a) (- b)"
 proof (rule sup_unique)
   fix a b c :: 'a
-  show "a \<le> - inf (-a) (-b)"
+  show "a \<le> - inf (- a) (- b)"
     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
       (simp, simp add: add_inf_distrib_left)
-  show "b \<le> - inf (-a) (-b)"
+  show "b \<le> - inf (- a) (- b)"
     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
       (simp, simp add: add_inf_distrib_left)
   assume "a \<le> c" "b \<le> c"
-  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
+  then show "- inf (- a) (- b) \<le> c"
+    by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
 qed
 
-lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
+lemma neg_inf_eq_sup: "- inf a b = sup (- a) (- b)"
   by (simp add: inf_eq_neg_sup)
 
 lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"
   using neg_inf_eq_sup [of b c, symmetric] by simp
 
-lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
+lemma neg_sup_eq_inf: "- sup a b = inf (- a) (- b)"
   by (simp add: sup_eq_neg_inf)
 
 lemma diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)"
@@ -95,13 +99,14 @@
 
 lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
 proof -
-  have "0 = - inf 0 (a-b) + inf (a-b) 0"
+  have "0 = - inf 0 (a - b) + inf (a - b) 0"
     by (simp add: inf_commute)
-  hence "0 = sup 0 (b-a) + inf (a-b) 0"
+  then have "0 = sup 0 (b - a) + inf (a - b) 0"
     by (simp add: inf_eq_neg_sup)
-  hence "0 = (-a + sup a b) + (inf a b + (-b))"
+  then have "0 = (- a + sup a b) + (inf a b + (- b))"
     by (simp only: add_sup_distrib_left add_inf_distrib_right) simp
-  then show ?thesis by (simp add: algebra_simps)
+  then show ?thesis
+    by (simp add: algebra_simps)
 qed
 
 
@@ -115,10 +120,13 @@
 
 lemma pprt_neg: "pprt (- x) = - nprt x"
 proof -
-  have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
-  also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
+  have "sup (- x) 0 = sup (- x) (- 0)"
+    unfolding minus_zero ..
+  also have "\<dots> = - inf x 0"
+    unfolding neg_inf_eq_sup ..
   finally have "sup (- x) 0 = - inf x 0" .
-  then show ?thesis unfolding pprt_def nprt_def .
+  then show ?thesis
+    unfolding pprt_def nprt_def .
 qed
 
 lemma nprt_neg: "nprt (- x) = - pprt x"
@@ -172,20 +180,26 @@
 lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
 proof -
   {
-    fix a::'a
-    assume hyp: "sup a (-a) = 0"
-    hence "sup a (-a) + a = a" by (simp)
-    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
-    hence "sup (a+a) 0 <= a" by (simp)
-    hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
+    fix a :: 'a
+    assume hyp: "sup a (- a) = 0"
+    then have "sup a (- a) + a = a"
+      by simp
+    then have "sup (a + a) 0 = a"
+      by (simp add: add_sup_distrib_right)
+    then have "sup (a + a) 0 \<le> a"
+      by simp
+    then have "0 \<le> a"
+      by (blast intro: order_trans inf_sup_ord)
   }
   note p = this
   assume hyp:"sup a (-a) = 0"
-  hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
-  from p[OF hyp] p[OF hyp2] show "a = 0" by simp
+  then have hyp2:"sup (-a) (-(-a)) = 0"
+    by (simp add: sup_commute)
+  from p[OF hyp] p[OF hyp2] show "a = 0"
+    by simp
 qed
 
-lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
+lemma inf_0_imp_0: "inf a (- a) = 0 \<Longrightarrow> a = 0"
   apply (simp add: inf_eq_neg_sup)
   apply (simp add: sup_commute)
   apply (erule sup_0_imp_0)
@@ -206,24 +220,32 @@
 lemma zero_le_double_add_iff_zero_le_single_add [simp]:
   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
 proof
-  assume "0 <= a + a"
-  hence a:"inf (a+a) 0 = 0" by (simp add: inf_commute inf_absorb1)
-  have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
+  assume "0 \<le> a + a"
+  then have a: "inf (a + a) 0 = 0"
+    by (simp add: inf_commute inf_absorb1)
+  have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l=_")
     by (simp add: add_sup_inf_distribs inf_aci)
-  hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
-  hence "inf a 0 = 0" by (simp only: add_right_cancel)
-  then show "0 <= a" unfolding le_iff_inf by (simp add: inf_commute)
+  then have "?l = 0 + inf a 0"
+    by (simp add: a, simp add: inf_commute)
+  then have "inf a 0 = 0"
+    by (simp only: add_right_cancel)
+  then show "0 \<le> a"
+    unfolding le_iff_inf by (simp add: inf_commute)
 next
-  assume a: "0 <= a"
-  show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
+  assume a: "0 \<le> a"
+  show "0 \<le> a + a"
+    by (simp add: add_mono[OF a a, simplified])
 qed
 
 lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
 proof
   assume assm: "a + a = 0"
-  then have "a + a + - a = - a" by simp
-  then have "a + (a + - a) = - a" by (simp only: add_assoc)
-  then have a: "- a = a" by simp
+  then have "a + a + - a = - a"
+    by simp
+  then have "a + (a + - a) = - a"
+    by (simp only: add_assoc)
+  then have a: "- a = a"
+    by simp
   show "a = 0"
     apply (rule antisym)
     apply (unfold neg_le_iff_le [symmetric, of a])
@@ -236,7 +258,8 @@
     done
 next
   assume "a = 0"
-  then show "a + a = 0" by simp
+  then show "a + a = 0"
+    by simp
 qed
 
 lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
@@ -261,19 +284,23 @@
 lemma double_add_le_zero_iff_single_add_le_zero [simp]:
   "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
 proof -
-  have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
+  have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)"
+    by (subst le_minus_iff, simp)
   moreover have "\<dots> \<longleftrightarrow> a \<le> 0"
     by (simp only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp
-  ultimately show ?thesis by blast
+  ultimately show ?thesis
+    by blast
 qed
 
 lemma double_add_less_zero_iff_single_less_zero [simp]:
   "a + a < 0 \<longleftrightarrow> a < 0"
 proof -
-  have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
+  have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"
+    by (subst less_minus_iff) simp
   moreover have "\<dots> \<longleftrightarrow> a < 0"
     by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp
-  ultimately show ?thesis by blast
+  ultimately show ?thesis
+    by blast
 qed
 
 declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] diff_inf_eq_sup [simp] diff_sup_eq_inf [simp]
@@ -281,17 +308,19 @@
 lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
 proof -
   from add_le_cancel_left [of "uminus a" "plus a a" zero]
-  have "(a <= -a) = (a+a <= 0)"
+  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0"
     by (simp add: add_assoc[symmetric])
-  thus ?thesis by simp
+  then show ?thesis
+    by simp
 qed
 
 lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
 proof -
   from add_le_cancel_left [of "uminus a" zero "plus a a"]
-  have "(-a <= a) = (0 <= a+a)"
+  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
     by (simp add: add_assoc[symmetric])
-  thus ?thesis by simp
+  then show ?thesis
+    by simp
 qed
 
 lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
@@ -314,7 +343,8 @@
 
 end
 
-lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
+lemmas add_sup_inf_distribs =
+  add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
 
 
 class lattice_ab_group_add_abs = lattice_ab_group_add + abs +
@@ -325,11 +355,15 @@
 proof -
   have "0 \<le> \<bar>a\<bar>"
   proof -
-    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
-    show ?thesis by (rule add_mono [OF a b, simplified])
+    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
+      by (auto simp add: abs_lattice)
+    show ?thesis
+      by (rule add_mono [OF a b, simplified])
   qed
-  then have "0 \<le> sup a (- a)" unfolding abs_lattice .
-  then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
+  then have "0 \<le> sup a (- a)"
+    unfolding abs_lattice .
+  then have "sup (sup a (- a)) 0 = sup a (- a)"
+    by (rule sup_absorb1)
   then show ?thesis
     by (simp add: add_sup_inf_distribs ac_simps pprt_def nprt_def abs_lattice)
 qed
@@ -347,7 +381,8 @@
   have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
     by (simp add: abs_lattice le_supI)
   fix a b
-  show "0 \<le> \<bar>a\<bar>" by simp
+  show "0 \<le> \<bar>a\<bar>"
+    by simp
   show "a \<le> \<bar>a\<bar>"
     by (auto simp add: abs_lattice)
   show "\<bar>-a\<bar> = \<bar>a\<bar>"
@@ -359,14 +394,20 @@
   }
   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
   proof -
-    have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
+    have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))"
+      (is "_=sup ?m ?n")
       by (simp add: abs_lattice add_sup_inf_distribs sup_aci ac_simps)
-    have a: "a + b <= sup ?m ?n" by simp
-    have b: "- a - b <= ?n" by simp
-    have c: "?n <= sup ?m ?n" by simp
-    from b c have d: "-a-b <= sup ?m ?n" by (rule order_trans)
-    have e:"-a-b = -(a+b)" by simp
-    from a d e have "abs(a+b) <= sup ?m ?n"
+    have a: "a + b \<le> sup ?m ?n"
+      by simp
+    have b: "- a - b \<le> ?n"
+      by simp
+    have c: "?n \<le> sup ?m ?n"
+      by simp
+    from b c have d: "- a - b \<le> sup ?m ?n"
+      by (rule order_trans)
+    have e: "- a - b = - (a + b)"
+      by simp
+    from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n"
       apply -
       apply (drule abs_leI)
       apply (simp_all only: algebra_simps ac_simps minus_add)
@@ -379,7 +420,7 @@
 end
 
 lemma sup_eq_if:
-  fixes a :: "'a\<Colon>{lattice_ab_group_add, linorder}"
+  fixes a :: "'a::{lattice_ab_group_add, linorder}"
   shows "sup a (- a) = (if a < 0 then - a else a)"
 proof -
   note add_le_cancel_right [of a a "- a", symmetric, simplified]
@@ -388,18 +429,23 @@
 qed
 
 lemma abs_if_lattice:
-  fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
+  fixes a :: "'a::{lattice_ab_group_add_abs, linorder}"
   shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
   by auto
 
 lemma estimate_by_abs:
-  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
+  fixes a b c :: "'a::lattice_ab_group_add_abs"
+  shows "a + b \<le> c \<Longrightarrow> a \<le> c + \<bar>b\<bar>"
 proof -
-  assume "a+b <= c"
-  then have "a <= c+(-b)" by (simp add: algebra_simps)
-  have "(-b) <= abs b" by (rule abs_ge_minus_self)
-  then have "c + (- b) \<le> c + \<bar>b\<bar>" by (rule add_left_mono)
-  with `a \<le> c + (- b)` show ?thesis by (rule order_trans)
+  assume "a + b \<le> c"
+  then have "a \<le> c + (- b)"
+    by (simp add: algebra_simps)
+  have "- b \<le> \<bar>b\<bar>"
+    by (rule abs_ge_minus_self)
+  then have "c + (- b) \<le> c + \<bar>b\<bar>"
+    by (rule add_left_mono)
+  with `a \<le> c + (- b)` show ?thesis
+    by (rule order_trans)
 qed
 
 class lattice_ring = ordered_ring + lattice_ab_group_add_abs
@@ -410,15 +456,17 @@
 
 end
 
-lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
+lemma abs_le_mult:
+  fixes a b :: "'a::lattice_ring"
+  shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>"
 proof -
   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
-  have a: "(abs a) * (abs b) = ?x"
+  have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x"
     by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
   {
     fix u v :: 'a
-    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
+    have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>
               u * v = pprt a * pprt b + pprt a * nprt b +
                       nprt a * pprt b + nprt a * nprt b"
       apply (subst prts[of u], subst prts[of v])
@@ -426,16 +474,22 @@
       done
   }
   note b = this[OF refl[of a] refl[of b]]
-  have xy: "- ?x <= ?y"
+  have xy: "- ?x \<le> ?y"
     apply simp
-    apply (metis (full_types) add_increasing add_uminus_conv_diff lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg mult_nonpos_nonpos nprt_le_zero zero_le_pprt)
+    apply (metis (full_types) add_increasing add_uminus_conv_diff
+      lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg
+      mult_nonpos_nonpos nprt_le_zero zero_le_pprt)
     done
-  have yx: "?y <= ?x"
+  have yx: "?y \<le> ?x"
     apply simp
-    apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos mult_nonpos_nonneg nprt_le_zero zero_le_pprt)
+    apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff
+      lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos
+      mult_nonpos_nonneg nprt_le_zero zero_le_pprt)
     done
-  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
-  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
+  have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"
+    by (simp only: a b yx)
+  have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"
+    by (simp only: a b xy)
   show ?thesis
     apply (rule abs_leI)
     apply (simp add: i1)
@@ -445,37 +499,38 @@
 
 instance lattice_ring \<subseteq> ordered_ring_abs
 proof
-  fix a b :: "'a\<Colon> lattice_ring"
+  fix a b :: "'a::lattice_ring"
   assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
-  show "abs (a*b) = abs a * abs b"
+  show "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
   proof -
-    have s: "(0 <= a*b) | (a*b <= 0)"
-      apply (auto)
+    have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)"
+      apply auto
       apply (rule_tac split_mult_pos_le)
-      apply (rule_tac contrapos_np[of "a*b <= 0"])
-      apply (simp)
+      apply (rule_tac contrapos_np[of "a * b \<le> 0"])
+      apply simp
       apply (rule_tac split_mult_neg_le)
-      apply (insert a)
-      apply (blast)
+      using a
+      apply blast
       done
     have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
       by (simp add: prts[symmetric])
     show ?thesis
-    proof cases
-      assume "0 <= a * b"
+    proof (cases "0 \<le> a * b")
+      case True
       then show ?thesis
         apply (simp_all add: mulprts abs_prts)
-        apply (insert a)
+        using a
         apply (auto simp add:
           algebra_simps
           iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
           iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
-          apply(drule (1) mult_nonneg_nonpos[of a b], simp)
-          apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
+        apply(drule (1) mult_nonneg_nonpos[of a b], simp)
+        apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
         done
     next
-      assume "~(0 <= a*b)"
-      with s have "a*b <= 0" by simp
+      case False
+      with s have "a * b \<le> 0"
+        by simp
       then show ?thesis
         apply (simp_all add: mulprts abs_prts)
         apply (insert a)
@@ -488,11 +543,12 @@
 qed
 
 lemma mult_le_prts:
-  assumes "a1 <= (a::'a::lattice_ring)"
-    and "a <= a2"
-    and "b1 <= b"
-    and "b <= b2"
-  shows "a * b <=
+  fixes a b :: "'a::lattice_ring"
+  assumes "a1 \<le> a"
+    and "a \<le> a2"
+    and "b1 \<le> b"
+    and "b \<le> b2"
+  shows "a * b \<le>
     pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
 proof -
   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
@@ -501,31 +557,31 @@
     done
   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
     by (simp add: algebra_simps)
-  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
+  moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"
     by (simp_all add: assms mult_mono)
-  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
+  moreover have "pprt a * nprt b \<le> pprt a1 * nprt b2"
   proof -
-    have "pprt a * nprt b <= pprt a * nprt b2"
+    have "pprt a * nprt b \<le> pprt a * nprt b2"
       by (simp add: mult_left_mono assms)
-    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
+    moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2"
       by (simp add: mult_right_mono_neg assms)
     ultimately show ?thesis
       by simp
   qed
-  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
+  moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1"
   proof -
-    have "nprt a * pprt b <= nprt a2 * pprt b"
+    have "nprt a * pprt b \<le> nprt a2 * pprt b"
       by (simp add: mult_right_mono assms)
-    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
+    moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1"
       by (simp add: mult_left_mono_neg assms)
     ultimately show ?thesis
       by simp
   qed
-  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
+  moreover have "nprt a * nprt b \<le> nprt a1 * nprt b1"
   proof -
-    have "nprt a * nprt b <= nprt a * nprt b1"
+    have "nprt a * nprt b \<le> nprt a * nprt b1"
       by (simp add: mult_left_mono_neg assms)
-    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
+    moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1"
       by (simp add: mult_right_mono_neg assms)
     ultimately show ?thesis
       by simp
@@ -537,36 +593,41 @@
 qed
 
 lemma mult_ge_prts:
-  assumes "a1 <= (a::'a::lattice_ring)"
-    and "a <= a2"
-    and "b1 <= b"
-    and "b <= b2"
-  shows "a * b >=
+  fixes a b :: "'a::lattice_ring"
+  assumes "a1 \<le> a"
+    and "a \<le> a2"
+    and "b1 \<le> b"
+    and "b \<le> b2"
+  shows "a * b \<ge>
     nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
 proof -
-  from assms have a1:"- a2 <= -a"
+  from assms have a1: "- a2 \<le> -a"
     by auto
-  from assms have a2: "-a <= -a1"
+  from assms have a2: "- a \<le> -a1"
     by auto
-  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
-  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
+  from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2",
+    OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
+  have le: "- (a * b) \<le> - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
+    - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
     by simp
-  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
+  then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
+      - pprt a1 * pprt b1 + - pprt a2 * nprt b1) \<le> a * b"
     by (simp only: minus_le_iff)
-  then show ?thesis by (simp add: algebra_simps)
+  then show ?thesis
+    by (simp add: algebra_simps)
 qed
 
 instance int :: lattice_ring
 proof
   fix k :: int
-  show "abs k = sup k (- k)"
+  show "\<bar>k\<bar> = sup k (- k)"
     by (auto simp add: sup_int_def)
 qed
 
 instance real :: lattice_ring
 proof
   fix a :: real
-  show "abs a = sup a (- a)"
+  show "\<bar>a\<bar> = sup a (- a)"
     by (auto simp add: sup_real_def)
 qed